Textbook Design Flaws May Be Behind Students’ Struggles With Linear Inequalities, Study Finds

When students fail algebra, the blame usually falls on the students themselves: they did not study hard enough, they lack aptitude, they were distracted. But a new study from Indonesia suggests that some of the deepest mathematical misunderstandings may be built into the very books children learn from. A team led by Anton Nasrullah of Universitas Bina Bangsa, working with colleagues at Universitas Pendidikan Indonesia, Nusa Putra University and INTI International University, examined how government-issued mathematics textbooks under Indonesia’s Merdeka Curriculum shape the way students learn systems of linear inequalities in two variables, a foundational algebra topic that underpins later subjects such as linear programming and optimization. Their conclusion is striking: the textbooks themselves appear to manufacture predictable learning obstacles, and the researchers can trace those obstacles directly into students’ exam papers.

The study, published in Discover Education, combined a close theoretical reading of the official textbooks with empirical testing of real students. The researchers analyzed both the teacher and student editions of the mandated textbooks using the Anthropological Theory of the Didactic, a framework developed by the French didactician Yves Chevallard that treats mathematical knowledge as a set of organized practices. Central to this framework is the concept of praxeology, the pairing of tasks and techniques with the explanations and theories that justify them. A well-designed textbook, in this view, does not merely show students how to graph an inequality; it explains why the shaded region represents an infinite set of solutions and why two overlapping regions must both be satisfied simultaneously. The Indonesian textbooks, the analysis found, repeatedly present the how while omitting the why.

That imbalance has a technical name in the literature: a praxeological imbalance, in which the practical block of tasks and techniques crowds out the logos block of justification and theory. The researchers documented concrete examples. In one opening problem, students weigh apples and oranges on a scale and are asked to build a system of inequalities through trial and error. The task is engaging, but it never explicitly instructs students to convert kilograms to ounces, a hidden technical demand that derails learners who have not mastered unit conversion. The completed mathematical model is then printed directly beside the problem, inviting imitation rather than independent reasoning. Elsewhere, the book leaps from single inequalities to full systems without scaffolding the crucial idea that a system’s solution is the intersection of two half-planes, not simply the area below each line.

To test whether these theoretical weaknesses translate into real difficulties, the team administered a diagnostic assessment of eight open-ended tasks to 80 students at a public vocational school in Banten, Indonesia, and then conducted in-depth interviews with 20 of them, selected purposively for showing distinctive error patterns. The tasks were deliberately mapped to the obstacle categories predicted by the textbook analysis: four targeted epistemological obstacles involving inequality symbols, solution sets and half-planes, three targeted didactical obstacles arising from instructional design, and one addressed ontogenic obstacles in applying the concepts to real-world contexts. The design meant that each error a student made could be traced back to a specific feature of how the material had been presented.

The results were remarkably consistent with the predictions. Half of the students, 50 percent, could draw boundary lines but could not interpret what the inequality symbols meant graphically, leaving them unable to decide which side of the line to shade. One student told the researchers, I have drawn the lines, but I don’t know whether to shade the upper or lower part. Twenty percent failed to identify the intersection of two regions as the solution set, instead shading everything below both lines; one admitted, I didn’t know I had to look for the intersection, I thought it was enough to shade below the lines. Another 30 percent chose test points randomly from the graph without substituting them into the inequalities, revealing a disconnect between graphical and symbolic representations that echoes Raymond Duval’s classic work on the cognitive difficulty of converting between mathematical registers.

The researchers also catalogued ontogenic obstacles, those rooted in students’ own developmental readiness. Conceptual gaps were the most common, affecting 55 percent of participants, who lacked command of prerequisites such as plotting lines and finding axis intercepts. Instrumental obstacles, meaning weak technical skills like substitution and arithmetic, appeared in 30 percent; one student wrote that 3 plus 3 equals 7 while checking a point against the inequality x plus y is at most 7. Psychological obstacles, including math anxiety and fear of failure, affected 15 percent, with students reporting they did not dare to try because they were afraid of being wrong. These affective barriers, the authors note, consume working memory resources and compound the conceptual difficulties, a finding consistent with the broader literature on cognitive load and mathematics anxiety.

Perhaps the most consequential category was the didactical obstacle, the kind produced by instruction itself. Forty-five percent of students shaded regions by pure imitation of textbook patterns, with one explaining, I shade the area below the line because that’s how it is in the book, but I’m not sure if it’s correct. Thirty-five percent had never been taught to seek the intersection of two constraints, and 20 percent showed no habit of algebraic validation at all, selecting points visually because, as one put it, the teacher usually never asks to check points. The researchers interpret these behaviors as products of an implicit didactical contract in which students see mathematics as a sequence of authoritative steps to be copied rather than a logical system requiring justification. When the textbook provides worked templates but no validation prompts, students learn that verification is not part of the job.

The team argues that these findings establish more than a correlation. By triangulating the textbook task layouts, the diagnostic error patterns and the interview testimonies, they built a mapping matrix linking specific textbook features to specific student errors. The high rate of shading mistakes corresponds to the books’ emphasis on mechanical table completion and the placement of finished models next to problems, which lowers cognitive demand and invites copying. The intersection and validation errors correspond to the textbooks’ silence on those very procedures. In the researchers’ framing, the obstacles are structurally embedded in the instructional material, not merely individual deficits, which means they can be anticipated and designed away.

The proposed remedies are concrete. The authors recommend a three-phase scaffolding model in which students first explore a contextual problem with guiding questions, then compare their reasoning with a partially completed model, and only finally see a fully worked example accompanied by explicit justification of each inequality. They call for standardized validation prompts after every graphing task, prerequisite activation boxes, and what-if prompts that ask how changing a strict inequality to an inclusive one alters the boundary. Comparative context strengthens the case: studies of Chinese textbooks by Liping Fan and Yan Zhu found more systematic scaffolding and explicit linking of models to tasks, suggesting the Indonesian materials lag behind international best practice. The researchers also urge a formal praxeological review of all government-published textbooks, so that every technique is paired with its conceptual rationale, and recommend that teachers supplement the books with think-aloud protocols and activities that make students justify their shading choices aloud.

The study’s limitations are acknowledged: it examined a single vocational school population and did not test an alternative textbook design longitudinally, so generalization requires caution. Yet its central message travels well beyond one Indonesian classroom. Textbooks are the primary, and often the only, reference in many mathematics classrooms worldwide, and any systematic weakness in their design propagates quietly through millions of learners. Systems of linear inequalities matter because they model budget constraints in economics and resource limits in engineering, and students who master them only mechanically tend to falter in linear programming, optimization and calculus. If the obstacles are built into the pages, the researchers conclude, then the fix belongs there too, in a deliberate redesign that treats conceptual understanding, solution validation and reflection not as extras but as the core of what a mathematics textbook is for.

Subject of Research: The impact of mathematics textbook design on learning obstacles and student performance in systems of linear inequalities in two variables

Article Title: Assessing the impact of mathematics textbooks on learning obstacles and student performance in systems of linear inequalities in two variables

Article References: Nasrullah, A., Suryadi, D., Prabawanto, S., Hendriyanto, A., Muhaimin, L. H., & Dewi, D. A. (2026). Assessing the impact of mathematics textbooks on learning obstacles and student performance in systems of linear inequalities in two variables. Discover Education, 5(1), Article 1036. https://doi.org/10.1007/s44217-026-02064-x

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02064-x

Keywords: mathematics education, textbook analysis, linear inequalities, learning obstacles, didactic transposition, praxeology, Merdeka Curriculum, algebra learning, cognitive load, math anxiety, scaffolding, Indonesia